3) At the start of the pandemic, there was a lot of talk about pooled testing...

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3) At the start of the pandemic, there was a lot of talk about pooled testing for covid as a cost effective and fast means of testing (there were no vaccines then and tests were not easily available; hence, being efficient in the number of tests being used was critical). The way pooled testing works is as follows: A random sample of, say, 5 people is selected, a nasal swab taken from each and then the material from the 5 swabs is pooled/combined and one single test is run on the pooled material. If this test comes back negative, all 5 people are declared negative. If this test comes back positive, then all the 5 individuals are tested separately (i.e. 5 tests). We will assume throughout this discussion that the test is a confirmatory test and not a diagnostic one (i.e. there are no false positives or false negatives) Suppose that 10% of the population in a city is infected (i.e. prob that person is infected is 0.1) and a pooled test is carried out on 5 randomly selected people. It can be shown, assuming that the event "a person is positive" is independent of "another person is positive", that the probability that the pooled test comes back positive is 0.41. (you can verify this for yourselves, as this question is an exact analogue of the Depeche Mode problem in HW 1) Suppose that one has a town of 50,000 people with an incidence rate of 0.1 (i.e. 10% of the population is infected). If the city decides to test every person individually, it would need to run 50,000 tests. However, if it breaks up the 50,000 individuals at random into 10,000 groups of 5 each, then it can run 10,000 pooled tests, though, each time a pooled test turns up positive, the city has to test each of the 5 people in that positive pool. a) Out of the 10,000 pooled tests, how many pooled tests would you expect to find positive? (HINT: Apply the long run relative frequency interpretation to the probability that a pooled test is positive, viz., 0.41). b) Based on this expected number of positive pooled tests, how many total tests (pooled and by individual) would the city expect to run? 3) At the start of the pandemic, there was a lot of talk about pooled testing for covid as a cost effective and fast means of testing (there were no vaccines then and tests were not easily available; hence, being efficient in the number of tests being used was critical). The way pooled testing works is as follows: A random sample of, say, 5 people is selected, a nasal swab taken from each and then the material from the 5 swabs is pooled/combined and one single test is run on the pooled material. If this test comes back negative, all 5 people are declared negative. If this test comes back positive, then all the 5 individuals are tested separately (i.e. 5 tests). We will assume throughout this discussion that the test is a confirmatory test and not a diagnostic one (i.e. there are no false positives or false negatives) Suppose that 10% of the population in a city is infected (i.e. prob that person is infected is 0.1) and a pooled test is carried out on 5 randomly selected people. It can be shown, assuming that the event "a person is positive" is independent of "another person is positive", that the probability that the pooled test comes back positive is 0.41. (you can verify this for yourselves, as this question is an exact analogue of the Depeche Mode problem in HW 1) Suppose that one has a town of 50,000 people with an incidence rate of 0.1 (i.e. 10% of the population is infected). If the city decides to test every person individually, it would need to run 50,000 tests. However, if it breaks up the 50,000 individuals at random into 10,000 groups of 5 each, then it can run 10,000 pooled tests, though, each time a pooled test turns up positive, the city has to test each of the 5 people in that positive pool. a) Out of the 10,000 pooled tests, how many pooled tests would you expect to find positive? (HINT: Apply the long run relative frequency interpretation to the probability that a pooled test is positive, viz., 0.41). b) Based on this expected number of positive pooled tests, how many total tests (pooled and by individual) would the city expect to run?